COMPOSITIONS and ORIGINS of OUTER PLANET SYSTEMS: INSIGHTS from the ROCHE CRITICAL DENSITY Matthew S

Draft version July 5, 2021 Preprint typeset using LATEX style emulateapj v. 5/2/11

COMPOSITIONS AND ORIGINS OF OUTER PLANET SYSTEMS: INSIGHTS FROM THE ROCHE CRITICAL DENSITY Matthew S. Tiscareno1, Matthew M. Hedman1, Joseph A. Burns2,3, and Julie Castillo-Rogez4 1Center for Radiophysics and Space Research, Cornell University, Ithaca, NY 14853, USA. 2Department of Astronomy, Cornell University, Ithaca, NY 14853, USA. 3College of Engineering, Cornell University, Ithaca, NY 14853, USA. 4Jet Propulsion Laboratory, Pasadena, CA 91109, USA.

Draft version July 5, 2021

ABSTRACT

We consider the Roche critical density (ρRoche), the minimum density of an orbiting object that, at a given distance from its planet, is able to hold itself together by self-gravity. It is directly related to the more familiar “Roche limit,” the distance from a planet at which a strengthless orbiting object of given density is pulled apart by tides. The presence of a substantial ring requires that transient clumps have an internal density less than ρRoche. Conversely, in the presence of abundant material for accretion, an orbiting object with density greater than ρRoche will grow. Comparing the ρRoche values at which the Saturn and Uranus systems transition rapidly from disruption-dominated (rings) to accretion-dominated (moons), we infer that the material composing Uranus’ rings is likely more rocky, as well as less porous, than that composing Saturn’s rings. From the high values of ρRoche at the innermost ring-moons of Jupiter and Neptune, we infer that those moons may be composed of denser material than expected, or more likely that they are interlopers that formed farther from their planets and have since migrated inward, now being held together by internal material strength. Finally, the “Portia group” of eight closely-packed Uranian moons has an overall surface density similar to that of Saturn’s A ring. Thus, it can be seen as an accretion-dominated ring system, of similar character to the standard ring systems except that its material has a characteristic density greater than the local ρRoche. Subject headings: Planets and satellites: composition — Planets and satellites: dynamical evolution and stability — Planets and satellites: formation — Planets and satellites: rings

1. THE ROCHE LIMIT AND THE where R is radius and ρ is internal density, and the sub- ROCHE CRITICAL DENSITY script “p” denotes the central planet. The “Roche limit” is the distance from a planet within The dimensionless geometrical parameter γ = 4π/3 ≈ which its tides can pull apart a strengthless compact ob- 4.2 for a sphere, but is smaller for an object that takes ject. Simply speaking, a ring∗ would be expected to re- a non-spherical shape with its long axis pointing toward side inside its planet’s Roche limit, while any disk of Saturn, as one would expect for an actively accreting material beyond that distance would be expected to ac- body and as observed for several of Saturn’s ring-moons crete into one or more moons. However, the Roche limit (Porco et al. 2007; Charnoz et al. 2007). The region of does not actually have a single value, but depends par- gravitational dominance of a point mass moon, known as ticularly on the density, rotation, and internal material the “Roche lobe” or the “Hill sphere,” is lemon-shaped, strength of the moon that may or may not get pulled with cusps at the two Lagrange points L1 and L2 (see, apart (Weidenschilling et al. 1984; Canup and Esposito e.g., Fig. 3.28 of Murray and Dermott 1999). Simply 1995). A simple value for the Roche limit of a spherical distributing the moon’s material into the shape of its arXiv:1302.1253v1 [astro-ph.EP] 6 Feb 2013 moon, assuming no internal strength, can be calculated Roche lobe, with uniform density, yields γ ≈ 1.6 (Porco from a balance between the tidal force (i.e., the differ- et al. 2007). Going further for the case of an incom- ence between the planet’s gravitational pull on one side pressible fluid, fully accounting for the feedback between of the moon and its pull on the other side) that would the moon’s distorted shape and its (now non-point-mass) tend to pull a moon apart, and the moon’s own grav- gravity field smooths out the cusps and further elongates ity that would tend to hold it together. This gives (e.g., the moon (Chandrasekhar 1969; Murray and Dermott Eq. 4.131 in Murray and Dermott 1999) 1999), leading to an even smaller value, γ ≈ 0.85. How- ever, some central mass concentration and the inability of a rubble pile with internal friction to exactly take its 4πρ 1/3 equilibrium shape will likely prevent γ from becoming a = R p , (1) Roche p γρ quite this low. Hereafter, we will use the Porco et al. (2007) value of γ ≈ 1.6. Because the moon’s internal density ρ appears in Eq. 1, ∗ Throughout this work, we use the word “ring” to refer to a substan- there is no single value of the Roche limit for a plane- tial annulus with sufficient material to support accretion. Our analysis tary system. This is intuitive, as a denser object can does not apply to tenuous structures such as Saturn’s G and E rings. 2

a simple clump of material with no core cannot increase its internal density by shedding material, and thus must disjoin completely if its ρ is less than ρRoche by a sufficient margin.† Therefore, the fundamental criterion determin- ing whether a particular location in a planetary system is characterized by disruption (rings) or accretion (dis- crete moons) is whether the density naturally achieved by transient clumps is greater or less than ρRoche. 2. APPLICATIONS 2.1. A clue to composition As explained in the previous section, the persistent existence of a ring at a given radial location a implies that the densities of transient clumps ρclump do not exceed ρRoche — that is, we expect that the composition of the ring material is such that ρclump . ρRoche. Because of Figure 1. The Roche critical density ρRoche (Eq. 2 or 3, with ‡ γ = 1.6) plotted against planetary radii for Jupiter (red), Saturn the porosity inherent in any transient clump, we can ex- (cyan), Uranus (green), and Neptune (blue). An object must have pect ρclump to be only a fraction of ρsolid, the density of a density higher than ρRoche to be held together by its own gravity; a solid chunk of the material that composes the ring. conversely, in the presence of abundant disk material, an embedded object will actively accrete as long as its density remains higher As seen in Fig. 1, Saturn’s main rings extend out- than ρRoche. The colored bars along the bottom show the extent ward to significantly lower values of ρRoche, approach- of each planet’s main ring system. The bars along the left-hand ing 0.4 g cm−3, than are seen in any of the other three side are constructed from the bars along the bottom simply by the known ring systems, probably reflecting their much lower latter reflecting off the appropriately colored diagonal line. For each bar, a solid circle indicates the outermost extent, and the rock fraction (and higher fraction of water ice) as already corresponding minimum ρRoche, of the main rings. Figure from known from spectroscopy and photometry. In fact, Sat- Tiscareno (2013). urn’s rings are composed almost entirely of water ice (Cuzzi et al. 2009), so we can conclude that ρclump is venture closer to the planet without danger of fragmen- ∼ 40% of ρsolid, at least in this case. Saturn’s “ring tation than can an object that is less dense (see Fig. 1 moons” from Pan through Pandora, which range from of Weidenschilling et al. 1984). In fact, it is often more 4 to 40 km in mean radius and orbit in the vicinity of useful in the context of rings to consider the limit from the rings, have similar densities and porosities (Thomas planetary tides not as a critical distance but rather as a 2010). critical density. The compositions of the Uranian and Neptunian rings We can rearrange Eq. 1 such that, at any given distance are almost entirely unknown, as Voyager did not carry an infrared spectrometer with enough spatial resolution a from the planet, the Roche critical density ρRoche at which the moon’s size entirely fills its Roche lobe is to detect them. However, it is clear from their low albedo that at least the surfaces of the ring particles cannot be 4πρp primarily water ice. Color imaging indicates that the ρRoche = 3 . (2) γ(a/Rp) Uranian rings are dark at all visible wavelengths, which would be consistent with the spectrum of carbon or or- 3 Then, substituting the planet’s mass Mp = (4π/3)Rpρp, ganics, among other possibilities. we have Like at Saturn, the Uranus system has a clear 3M ρ = p . (3) boundary between disruption-dominated and accretion- Roche γa3 dominated regions (Fig. 2), notwithstanding some minor mixing at the boundary — at Uranus, Cordelia is in- While Eq. 3 gives the simplest method for calculating ward of the ring; at Saturn, Pan and Daphnis orbit ρRoche, Eq. 2 expresses its dependence on the distance in within the outermost parts of the A ring. The Uranian terms of planetary radii (a/Rp); when shown on a log-log § −3 main rings extend only to ρRoche ≈ 1.2 g cm . If we plot, as in Fig. 1, it appears as a straight line of slope assume a porosity for Uranian ring material with a sim- −3, normalized by each planet’s internal density ρp. ilar value (∼ 40%) as at Saturn, then a solid chunk of Within a ring, where material for accretion is plenti- the material that composes the ring could be as high ful and collisions are relatively gentle, any pre-existing −3 as ρsolid ≈ 3 g cm . However, it is probably not that solid chunk with internal density ρ > ρRoche should ac- high, as that value would imply a composition almost en- crete a mantle of porous ring material until its density decreases to match ρRoche (which is to say, the resulting † Even a rubble pile has some internal material strength induced by entity fills its Roche lobe). Indeed, the moons near and granular friction, which can serve to hold it together in the face of tidal forces that would otherwise tear it apart. Therefore, the actual critical within Saturn’s rings are observed to have ρ ∼ ρRoche and density for breakup is somewhat less than ρRoche (e.g., Sharma 2009). to have shapes reminiscent of their Roche lobes (Porco In this work, however, we assume that this effect is small and do not et al. 2007; Charnoz et al. 2007). On the other hand, treat it quantitatively. any loose agglomeration of material with internal den- ‡ Recall that the volume fraction of close-packed equal spheres is only ∼ 70%. sity ρ < ρRoche must shed material. Such an object may § The limiting value of ρRoche for the Uranus system may be as high be the aforementioned solid core with an icy mantle, if as 1.4 g cm−3 (the value for the δ ring) if the highly perturbed state it has grown beyond its Roche lobe, in which case it will of the ring increases its resistance to accretion, a question that has simply shrink back to the size of its Roche lobe. However, not been studied in detail. 3

Figure 2. Schematic of the four outer planet ring systems, with a common scale based on ρRoche. Shades of gray indicate ring surface densities, with black indicating regions that are empty (so far as is known) and white indicating the highest surface densities (or the planet itself). The red line in each panel indicates the location where the orbit rate is synchronous with the planet’s rotation. In the Uranus panel, the arrow indicates the extent of the “Portia group” of moons, as discussed in the text. Figure from Hedman (2013). tirely of rock, while lower-density materials (such as ices, cannot use the location of Neptune’s rings to provide a clathrates, and perhaps organics) are likely to be present reliable constraint on the ring material’s composition or in signiﬁcant amounts. Nonetheless, we can safely con- porosity. clude from this observation that the Uranian rings are The extent of Jupiter’s Main ring, in contrast to the likely composed of material with a much higher rock frac- other three ring systems, is clearly limited by the avail- tion than at Saturn, as well as that the porosity of tran- ability of material (which, other than dust, is restricted sient clumps is somewhat less than at Saturn. to the narrow region between the source moons Metis Our inference from the Roche critical density at the and Adrastea, and which is not abundant) rather than ring/moon transition, that the Uranus system is rock- by a disruption/accretion balance. However, Jupiter’s −3 ier overall than the Saturn system, is consistent with high transitional ρRoche ≈ 1.7 g cm places the only the fact that the average density of Saturn’s mid-size known limit on the densities (and thus masses) of the −3 moons (Matson et al. 2009) is 1.2 g cm , while that of source moons, which must be denser than ρRoche in or- Uranus’ major moons (Jacobson et al. 1992) is 1.6 g cm−3 der to hold themselves together by gravity. However, it (although the lower density of Uranus’ innermost major may not be valid to assume that Metis and Adrastea are moon, Miranda at 1.2 ± 0.15 g/cm3, may hint at hetero- held together by gravity, as accreting masses must be, geneous accretion within the protosatellite disk). Fur- given the large gap in particle size between the ∼ 10-km thermore, the 40% increase in moon density from Saturn moons and other Main ring particles, which observation- to Uranus, rather than a threefold increase as for the ally cannot be larger than 1 km (Showalter et al. 2007). transitional ρRoche, supports the inference that porosity This large gap in particle size might be explained if Metis is greater in Saturn’s ice-rich ring material. and Adrastea are solid bodies originating further from The outer edge of Neptune’s main ring system corre- Jupiter, now held together by material strength, while −3 sponds to ρRoche ≈ 0.8 g cm . This is intermediate no bodies of similar size are now able to form through between the values for the Saturn and Uranus systems, in situ accretion. possibly indicating an intermediate rock/ice ratio. How- 2.2. A clue to past dynamics ever, Neptune’s rings are much less substantial than Sat- urn’s or Uranus’ (its highest optical depths are compa- Unlike Saturn and Uranus, Jupiter and Neptune do rable to those of the C ring and the Cassini Division, not have clear boundaries between accretion-dominated but with considerably more dust), and even the Adams and disruption-dominated regions. Both have moons in- ring may be nothing more than an assemblage of dust- terspersed with their rings, down to values of ρRoche as shedding moonlets. Furthermore, any indigenous major high as 2 g cm−3 (Fig. 2). In the case of Jupiter, the moons of Neptune were likely lost during Triton’s cap- value of ρRoche in the vicinity of the innermost moons is ture (Goldreich et al. 1989; Agnor and Hamilton 2006), twice as high as the measured bulk density of Amalthea so we have no independent way to estimate the compo- (Anderson et al. 2005). In the case of Neptune, it is three sition of Neptune’s small moons or rings. We therefore times as high as the constraint placed on the densities of the inner moons by Zhang and Hamilton (2007, 2008) in 4

Table 1 occupying an annulus from 2.31 to 2.99 RU (Duncan and Internal strength required to hold together the innermost known Levison 1997; Showalter and Lissauer 2006). The orbits moon in each of the giant planet systems, if ρ = 1 g cm−3. of the moons in this group are dynamically unstable on

−3 timescales of ∼ 10 Myr (Showalter and Lissauer 2006; Name (System) R (km) ρRoche (g cm )σ ¯ (kPa) Dawson et al. 2010; French and Showalter 2012), and Metis (Jupiter) 22 1.71 19.2 Pan (Saturn) 14 0.45 — the region has likely looked quite different over solar sys- Cordelia (Uranus) 20 1.32 7.2 tem history as moons are disrupted and re-accreted on Naiad (Neptune) 33 1.71 43.4 a regular basis. The dusty ν ring, which lies between two of the moons of the Portia group, may indeed be the detritus of a recent significant collision, perhaps the order to account for their low inclinations. The existence disruption of a moon. French and Showalter (2012) have suggested that, some 10 Myr in the future, it is likely in both systems of moons at such high values of ρRoche may indicate that they are interlopers that formed far- that the ν ring will have re-accreted into a new moon, ther from their planets and have since migrated inward while one or more current moons (Cupid seems a likely and are held together by internal material strength. Al- candidate) will have been destroyed and will temporarily ternatively, it may indicate that they are composed of look as the ν ring does now. However, detailed study of denser (likely silicate) material. As the internal strength this scenario remains to be carried out. required to hold together a moon is given by (Holsapple The mean surface density of the Portia group region, and Michel 2008) calculated by spreading the moons’ mass evenly over the annulus containing them, is ∼ 45 g cm−2, comparable to 4πR2Gρ the surface density of Saturn’s A ring (Tiscareno et al. σ¯ = (ρ − ρ) , (4) 15 Roche 2007). With all this in mind, Uranus’ Portia group can be seen as being of similar character to the known dense ring our analysis indicates that all of the known moons in systems, with the difference being that it is dominated the outer solar system can be held together with internal by accretion rather than by disruption, due to the low strengths of less than 50 kPa (Table 1), which is several value of ρ at its location. orders of magnitude smaller than typical strengths for Roche ice-rock mixtures (Durham et al. 1992). 3. CONCLUSIONS In the case of Neptune, the icy composition of most We consider the Roche critical density (ρRoche), the objects that far from the Sun suggests that its moons minimum density of an orbiting object that, at a given have significant internal strength, though it is possible distance from its planet, is able to hold itself together for Neptune’s moons to be more silicate-rich than its by self-gravity. It is directly related to the more familiar rings (as is, in fact, the case for Saturn’s mid-size moons, “Roche limit,” the distance from a planet at which an though not for Saturn’s ring-moons). In the case of orbiting object of given density is pulled apart by tides. Jupiter, the importance of internal strength is corrob- At a given distance from the planet, an orbiting object orated by New Horizons’s non-detection of objects with whose density is less than ρRoche will be pulled apart sizes intermediate between Metis and Adrastea and the (unless it has internal material strength to hold itself to- continuum ring particles (Showalter et al. 2007). gether), and its material will form a ring. Conversely, The hypothesis that the inner moons of Jupiter and in the presence of abundant material for accretion, an Neptune may have formed farther from their planets orbiting object whose density is greater than ρRoche will may even include possible formation elsewhere in the so- accrete material; if the accreted material is more fluffy lar system. Charnoz et al. (2009) found that Jupiter than the dense core, the object’s bulk density may de- and Neptune are likely to have received the most ma- crease until equal to ρRoche, at which point it will have terial from circum-solar orbits during the Late Heavy “filled its Roche zone” and will stop accreting. Bombardment; therefore, those two planets are the most Both Saturn and Uranus have relatively clear bound- likely to have captured an object from heliocentric orbit. aries between accretion-dominated regions (populated Capture could be facilitated near the Roche limit (given with moons) and disruption-dominated regions (popu- the captured object’s density) as internal dissipation is lated with rings). Given the presence, in both cases, of increased for an object near breakup. abundant material for accretion, the value of ρRoche at However, a gentler scenario is available, in that the the boundary should be slightly higher than the density inner moons of both Neptune and Jupiter (as well as of transient clumps. Noting that the boundary value Uranus) are well inward of synchronous orbit (Fig. 2). at Uranus is three times higher than at Saturn, we in- So they may have formed in the latter vicinity, where fer that the material composing Uranus’ rings is likely ρRoche is relatively low, and subsequently migrated in- more rocky, as well as less porous, than that composing ward under tidal evolution. Dynamical constraints on Saturn’s rings. This inference is consistent with several how much Io can have migrated over its history (Hamil- observational clues. ton 2011) should not limit the movement of Jupiter’s in- Neptune and Jupiter have moons and rings inter- ner moons to their current locations through gradual pro- spersed, with moons at values of ρRoche as high as cesses (D. P. Hamilton, personal communication, 2012). 2 g cm−3. This may indicate that the moons are interlopers that formed farther from their planets and have since 2.3. An accretion-dominated ring migrated inward and are held together by internal mate- Uranus’ inner complement of moons (which we will call rial strength, or it may indicate that they are composed the “Portia group” after its leading member) is the most of denser (likely silicate) material. densely-packed known satellite system, with eight moons The “Portia group” of eight Uranian moons packed 5 between 2.31 to 2.99 RU has an overall surface density Hedman, M. M. 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